Question

Consider a sample with a mean of 30 and a standard deviation of 6. Use Chebyshev's theorem to determine the percentage of the data within each of the following ranges(to the nearest whole number).

a) 10 to 50, at least ____%

b)15 to 45, at least ___%

c) 22 to 38, at least ___%

d) 17 to 43, at least ___%

e) 14 to 46, at least ___%

Answer 1

Chebyshev's theorem:-

SD = 6

M= 30

CASE 1 ( 10 TO 50 )

K= UL - M/SD = 50 - 30/6 = 3.33

1- 1/k² = 1-1/3.33²= 1-1/11.0889 = 11.0889 -1/11.0889 = 10.0889/11.0889 = 0.90 =90 %

CASE 2 (15 TO 45)

K = UL -M /SD = 45 - 30 / 6 = 2.5

1-1/K² = 1- 1/ 2.5² = 1-1/6.25 = 6.25 - 1 / 6.25 = 5.25 /6.25 = 0.84 = 84%

CASE 3 (22 TO 38 )

K= UL - M / SD = 38 - 30/6 = 1.33

1-1/K² = 1 - 1/ 1.33²= 1- 1/ 1.7778 =0.4375=43.75%

CASE 4 (17 TO 43)

K = UL -M / SD = 43 - 30 / 6 = 2.16667

1 - 1/ K² = 1- 1/ 2.16667² = 0.786983=78.70%

CASE 5 ( 14 TO 46)

K= UL - M / SD = 46 - 30 / 6 = 2.66667

1 - 1 / K² = 1 - 1 / 2.66667² = 0.8594=85.94%